So, here we end up with 14 field correction coefficients (\(\nu _{1,2,3},\epsilon _{1,2,3},\pi _{1,2,3},\theta _{1,3},\gamma _{1,3}\)  and the shear viscosity \(\eta \) ). It was shown in [1]}, [2]}
that not all coefficients are invariant under the first-order field redefinition
(due to the arbitrariness in the definition of temperature, fluid four-velocity and chemical potential for out-of equilibrium case). We checked that our coefficients satisfy that the combinations
\(f_i=\pi _i-\varepsilon _i\left(\frac{\partial P_0}{\partial \epsilon _0}\right)_{n_0}-\nu _i \left(\frac{\partial P_0}{\partial n_0}\right)_{\epsilon _0}\)  and
\(l_i=\gamma _i-\frac{n_0}{\epsilon _0+P_0}\theta _i\)  are frame invariant (i.e., independent of the indices \(i,j,k\) ), which further reduce to the physical transport coefficients; bulk viscosity
\(\zeta =-f_2+\left(\frac{\partial P_0}{\partial \epsilon _0}\right)_{n_0}f_1+\frac{1}{T}\left(\frac{\partial P_0}{\partial n_0}\right)_{\epsilon _0}f_3\) , and charge conductivity
\(k_n=l_3-\frac{n_0T}{(\epsilon _0+P_0)}l_1\) . The detailed expressions of \(\zeta \)  and \(k_n\)  with MDRTA are given in [3]}. The corrections further reveal that LL and Eckart limit of the scalar
index (\(i=1,j=2\)  or vice versa) give \(\delta n^{(1)}=0,\delta \epsilon ^{(1)}=0\)  (such that \(\zeta \)  is entirety taken up by pressure correction), where for the vector index LL limit (\(k=1\) ) gives \(W^{(1)\mu }=0\)  and Eckart limit (\(k=0\) ) gives \(V^{(1)\mu }=0\) .
Most significantly, we found that for the momentum independent relaxation time (i.e., for \(\Lambda \) =0) all the correction coefficients associated with the first-order time derivatives
\((\nu _1,\nu _3,\varepsilon _1,\varepsilon _3,\pi _1,\pi _3,\theta _1,\gamma _1)\)  in Eqs. (REF )-() identically vanish for all hydrodynamic frame conditions (irrespective of \(i,j,k\)  values)
which will be shown later to have crucial implications on the causality and stability of the theory.
